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I'm learning space group, as an exercise, I'm trying to determine all the 14 Bravais lattices by adding base-centered, body-centered or face-centered atom on the simple lattice of 7 crystal family.

But when I'm confronted with hexagonal crystal family, I can't find out why there can't be base-centered, body-centered or face-centered rhombohedral/hexagonal lattice.

Hope someone can show me how to argue there can't be. (for both rhombohedral and hexagonal)

graphics will be of great help!

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Fast answer: you can have these centered lattices... they are just not useful. Either they are incompatible with symmetry or they can be reduced to smaller cells.

Imagine/draw a hexagonal C face-centered cell. The resulting lattice is incompatible with the 6-fold symmetry. (For the same reason, you cannot have cubic face-centered. Only cubic all-faced centered. To make the C centering compatible with 6-fold symmetry, you have to add additional lattice points in the middle of the a and b axes. But if you do that you can just use a smaller, primitive hexagonal cell to describe the same lattice.

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